3.256 \(\int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\)

Optimal. Leaf size=62 \[ \frac {2}{3} x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a c x}+\frac {8 c x \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-a c x}} \]

[Out]

8/3*c*x*(1-1/a^2/x^2)^(1/2)/(-a*c*x+c)^(1/2)+2/3*x*(1-1/a^2/x^2)^(1/2)*(-a*c*x+c)^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 89, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6176, 6181, 78, 37} \[ \frac {2 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {10 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Int[Sqrt[c - a*c*x]/E^ArcCoth[a*x],x]

[Out]

(-10*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(3*S
qrt[1 - 1/(a*x)])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 6176

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n,
p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int e^{-\coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} \sqrt {x} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x}{a}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {\left (5 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 a \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {10 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 0.81 \[ \frac {2 \sqrt {\frac {1}{a x}+1} (a x-5) \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a*c*x]/E^ArcCoth[a*x],x]

[Out]

(2*Sqrt[1 + 1/(a*x)]*(-5 + a*x)*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])

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fricas [A]  time = 0.58, size = 50, normalized size = 0.81 \[ \frac {2 \, {\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} x - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

2/3*(a^2*x^2 - 4*a*x - 5)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)

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giac [A]  time = 0.14, size = 43, normalized size = 0.69 \[ \frac {2 \, {\left (-a c x - c\right )}^{\frac {3}{2}} {\left | c \right |}}{3 \, a c^{2}} + \frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

2/3*(-a*c*x - c)^(3/2)*abs(c)/(a*c^2) + 4*sqrt(-a*c*x - c)*abs(c)/(a*c)

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maple [A]  time = 0.04, size = 47, normalized size = 0.76 \[ \frac {2 \left (a x +1\right ) \left (a x -5\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{3 \left (a x -1\right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x)

[Out]

2/3*(a*x+1)*(a*x-5)*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)/a

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maxima [A]  time = 0.33, size = 54, normalized size = 0.87 \[ \frac {2 \, {\left (a^{2} \sqrt {-c} x^{2} - 4 \, a \sqrt {-c} x - 5 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{3 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

2/3*(a^2*sqrt(-c)*x^2 - 4*a*sqrt(-c)*x - 5*sqrt(-c))*(a*x - 1)/((a^2*x - a)*sqrt(a*x + 1))

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mupad [B]  time = 1.28, size = 71, normalized size = 1.15 \[ \frac {2\,\sqrt {c-a\,c\,x}\,\left (a\,x-3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a}-\frac {16\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,a\,\left (a\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

(2*(c - a*c*x)^(1/2)*(a*x - 3)*((a*x - 1)/(a*x + 1))^(1/2))/(3*a) - (16*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1)
)^(1/2))/(3*a*(a*x - 1))

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sympy [C]  time = 42.84, size = 66, normalized size = 1.06 \[ \frac {4 i c x \sqrt {\frac {1}{a c x + c}}}{3} + \frac {4 i c \sqrt {\frac {1}{a c x + c}}}{a} - \frac {2 i \left (- a c x + c\right )^{2} \sqrt {\frac {1}{a c x + c}}}{3 a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

4*I*c*x*sqrt(1/(a*c*x + c))/3 + 4*I*c*sqrt(1/(a*c*x + c))/a - 2*I*(-a*c*x + c)**2*sqrt(1/(a*c*x + c))/(3*a*c)

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